I think you're oversimplifying the scenario by implying that US assets (I have GPS in mind for instance) could only suffer "cosmetic" alterations in operation. Assets to counter satellite operations have been developing for a long time in Russia: hunter satellites, high power lasers, rockets (especially rockets - I would think Russian Space Forces wouldn't have much trouble with targeting and launching missiles fairly quickly - not an overly complicated task for a sophisticated space program).
Using the GPS satellite constellation as an example, there are 27 satellites in the constellation, of which I believe 24 are 'online' with the other three being spares available to take over should one or more of the online satellites fail. The GPS satellites also orbit the Earth at a distance of 12,000 miles fast enough to complete a rotation in 12 hours. The orbital tracks are also positioned so that at any given point in time a GPS receiver on Earth can get telemetry from 4 different positioning satellites.
In order to have any impact on the GPS system, at least 4 different satellites need to be effected. Unless the hypothetical conflict was truly worldwide, the GPS satellite constellation might be able to be re-arranged to just provide coverage of conflict areas until replacement satellites could be launched. This would then mean that a greater number of GPS satellites would need to be effected to begin degrading their performance over the conflict area.
Given the numbers and the velocities and distances involved, I do not believe any nation is capable of effecting sufficient GPS satellites to impact the system. Part of that belief is that it is harder to get an asset into space and then bring it close enough to an already maneuvering satellite so that it can have an effect upon the satellite. The other reason for that belief is that GPS started as a military/defence effort to improve navigation.
One concern during development and improvement (work is currently underway for GPS-IIF satellites and GPS-III) is that attacks could be launched on the satellites in an effort to bring down the system. With that in mind, I would expect that efforts would have been made to 'harden' the GPS satellites to make them further resistant to EM effects, beyond what is required normally in space. Another way to counter efforts to cripple the system would be to just increase the number of GPS satellites in order. While not necessarily an inexpensive solution, if is believed that an opponent can successfully persecute an attack upon GPS satellites why not just increase the numbers needed to be targeted before system failure occurs?
By my admittedly very rough guestimates, if the desired impact was a loss of GPS coverage to 25% of the world, then 10 GPS satellites would need to be effected. If there were 48 GPS satellites orbiting instead of 27, then something like 30 satellites would need to be effected...
This whole scenario also ignores the fact that alternate methods of guidance and navigation are available as adjuncts to, or in place of GPS navigation. This means that even with the loss of GPS, it only leads to a degradation in accuracy of navigation, not a loss in ability to navigate.
As Waylander put, a US/NATO non-nuclear conflict with an advanced nation would likely mean that US & NATO forces would encounter equipment and tactics which would degrade their operational effectiveness more so than if the conflict was not with a peer or near-peer nation. This would in turn means that the US/NATO forces would likely suffer higher losses, and/or need to expend more equipment/munitions/time in order to achieve victory.
So what?
By using nations X and Y I was attempting to keep this from being a direct comparison between various nations and therefore not just another Russia vs. US/NATO. If that is what people want, consider the following.
For approximately forty years, the US and NATO had made plans and developed equipment to keep Soviet & Warsaw Pact forces from crossing the Fulda Gap. At the time this was going on, the Soviet Union was a peer nation to the US in terms of influence, tech development, defence forces, etc. In the last two decades, the Soviet Union has broken up into a number of separate states, not all of which are friendly with each other. The Warsaw Pact, a defence agreement meant to oppose NATO has also ended, and a number of former member-nations are now friendly with, or even members of NATO. Following the breakup and the subsequent economic upheaval in the former Soviet states, there was a marked decrease in terms of development and production of defence equipment. There were also cuts in the level of forces maintained, with some capabilities being severely curtailed or even given up entirely.
In short, the Russia of today lacks much of the capability that the Soviet Union of 25 years had. When compared with peer nations which have been able to maintain various capabilities and improve upon them and develop new capabilities, the difference is even more noticeable.
-Cheers